Sums of two squares and the tau-function: Ramanujan's trail
Bruce C. Berndt, Pieter Moree
备注：
survey article, 38 pages, 3 tables, 208 references. We took around 30 suggestions of the two referees into account. To appear in Expositiones Mathematicae
Ramanujan, in his famous first letter to Hardy, claimed a very precise estimate for the number of integers that can be written as a sum of two squares. Far less well-known is that he also made further claims of a similar nature for the non-divisibility of the Ramanujan tau-function for certain primes. In this survey, we provide more historical details and also discuss related later developments. These show that, as so often, Ramanujan was an explorer in a fascinating wilderness, leaving behind him a beckoning trail.
In this paper we attempt to prove Lehmer's conjecture on Ramanujan's tau function, namely tau(n) is never zero, for each n larger than zero by investigating the additive group structure attached to tau(n) with the aid of unique factorization theorem.
We use Rankin-Cohen brackets for modular forms and quasimodular forms to give a different proof of the results obtained by D. Lanphier and D. Niebur on the van der Pol type identities for the Ramanujan's tau function. As consequences we obtain convolution sums and congruence relations involving the divisor functions.
We show that if $E/\mathbb{Q}$ is an elliptic curve with a rational $p$-torsion for $p=2$ or $3$, then there is a congruence relation between Ramanujan's tau function and $E$ modulo $p$. We make use of such congruences to compute the Iwasawa invariants of $2$-adic and $3$-adic Mazur--Tate elements attached to Ramanujan's tau function. We also investigate numerically the Iwasawa invariants of the Mazur--Tate elements attached to an elliptic curve with additive reduction at a fixed prime number.
We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for $n>1$ we prove that $$τ(n)\not \in \{\pm 1, \pm 3, \pm 5, \pm 7, \pm 691\}.$$ This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of the authors with Wei-Lun Tsai. Ramanujan's well-known congruences for $τ(n)$ allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty-Coleman method for hyperelliptic curves, and facts about ce…
Given a commutative ring $R$ with identity, let $H_R$ be the set of sequences of elements in $R$. We investigate a novel isomorphism between $(H_R, +)$ and $(\tilde H_R,*)$, where $+$ is the componentwise sum, $*$ is the convolution product (or Cauchy product) and $\tilde H_R$ the set of sequences starting with $1_R$. We also define a recursive transform over $H_R$ that, together to the isomorphism, allows to highlight new relations among some well studied integer sequences. Moreover, these connections allow to introduce a family of polynomials connected to the D'Arcais numbers and the Ramanuj…
We use a q-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan's tau function which involves t-cores and a new class of partitions which we call (m,k)-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.
In this paper, we find the number of representations of the quadratic form $x_1^2+ x_1x_2 + x_2^2 + \ldots + x_{2k-1}^2 + x_{2k-1}x_{2k} + x_{2k}^2,$ for $k=7,9,11,12,14$ using the theory of modular forms. By comparing our formulas with the formulas obtained by G. A. Lomadze, we obtain the Fourier coefficients of certain newforms of level $3$ and weights $7,9,11$ in terms of certain finite sums involving the solutions of similar quadratic forms of lower variables. In the case of $24$ variables, comparison of these formulas gives rise to a new formula for the Ramanujan Tau function.
Towards the end of his life Ramanujan wrote a manuscript on properties of the partition and tau functions, some parts of which remained unpublished until very recently. Nevertheless, this manuscript gave rise to a lot of subsequent work. In it Ramanujan considers congruences for $τ(n)$ modulo some special primes q. He proves for example that $τ(n)\equiv \sum_{d|n}d^{11}({\rm mod}691)$. He defines $t_n=1$ if $τ(n)$ is not divisible by q and $t_n=0$ otherwise. He then typically writes: "It can be shown by transcendental methods that $$ \sum_{k=1}^n t_k=C\int_1^n {dx\over (\log x)^δ}+O({n\over (\…
In the spirit of Lehmer's speculation that Ramanujan's tau-function never vanishes, it is natural to ask whether any given integer $α$ is a value of $τ(n)$. For odd $α$, Murty, Murty, and Shorey proved that $τ(n)\neq α$ for sufficiently large $n$. Several recent papers have identified explicit examples of odd $α$ which are not tau-values. Here we apply these results (most notably the recent work of Bennett, Gherga, Patel, and Siksek) to offer the first examples of even integers that are not tau-values. Namely, for primes $\ell$ we find that $$ τ(n)\not \in \{ \pm 2\ell \ : \ 3\leq \ell< 100\} …
We prove a number of results regarding odd values of the Ramanujan $τ$-function. For example, we prove the existence of an effectively computable positive constant $κ$ such that if $τ(n)$ is odd and $n \ge 25$ then either \[ P(τ(n)) \; > \; κ\cdot \frac{\log\log\log{n}}{\log\log\log\log{n}} \] or there exists a prime $p \mid n$ with $τ(p)=0$. Here $P(m)$ denotes the largest prime factor of $m$. We also solve the equation $τ(n)=\pm 3^{b_1} 5^{b_2} 7^{b_3} 11^{b_4}$ and the equations $τ(n)=\pm q^b$ where $3\le q < 100$ is prime and the exponents are arbitrary nonnegative integers. We make use of…
This paper aims to introduce two systems of nonlinear ordinary differential equations whose solution components generate the graded algebra of quasi-modular forms on Hecke congruence subgroups $Γ_0(2)$ and $Γ_0(3)$. Using these systems, we provide the generated graded algebras with an $\mathfrak{sl}_2(\mathbb{C})$-module structure. As applications, we introduce Ramanujan-type tau functions for $Γ_0(2)$ and $Γ_0(3)$, and obtain some interesting and non-trivial recurrence and congruence relations.
In this paper we prove Lehmer's conjecture on Ramanujan's tau function, namely tau(n) not equal to zero for n >= 1 by investigating the additive group structure attached to tau(n) with the aid of the pigeonhole principle and unique factorization theorem.
We prove formulas for special values of the Ramanujan tau zeta function. Our formulas show that $L(Δ, k)$ is a period in the sense of Kontsevich and Zagier when $k\ge12$. As an illustration, we reduce $L(Δ, k)$ to explicit integrals of hypergeometric and algebraic functions when $k\in\{12,13,14,15\}$.
We propose an improved algorithm for computing mod $\ell$ Galois representations associated to a cusp form $f$ of level one. The proposed method allows us to explicitly compute the case with $\ell=29$ and $f$ of weight $k=16$, and the cases with $\ell=31$ and $f$ of weight $k=12,20, 22$. All the results are rigorously proved to be correct. As an example, we will compute the values modulo $31$ of Ramanujan's tau function at some huge primes up to a sign. Also we will give an improved higher bound on Lehmer's conjecture for Ramanujan's tau function.
When Mike Hirschhorn showed us his lovely gem, that gives the simplest-to-date proof of Ramanujan's famous result that p(11n+6) is divisible by 11, we realized that his amazing method can be extended, and taught to a computer, and can prove even deeper identities. We would have done much more if not for the existence of Silviu Radu's powerful algorithm that handle any Ramanujan type congruence for any modular form (of a very general type), but it is still nice to know that in order to prove such simply stated results, that can be explained to a seven-year-old, one does not need the intimidatin…
In this paper, we give a generate function for the $σ_1$ function. Then we find some connections between the $σ_1$ function and the Ramanujan's tau function. We hope this connection will give some insights into the unsolved problems in classical number theory.
We consider very general "random integers" and (attempt to) prove that many multiplicative and additive functions of such integers have limiting distributions. These integers include, for instance, the curvatures of Apollonian circle packings, trace of Frobenius elements for elliptic curves, the Ramanujan tau-function, Mersenne numbers, and many others.
Hafner and Stopple proved a conjecture of Zagier, that the inverse Mellin transform of the symmetric square $L$-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the non-trivial zeros of the Riemann zeta function $ζ(s)$. Later, Chakraborty, Kanemitsu and the second author extended this phenomenon for any Hecke eigenform over the full modular group. In this paper, we study an asymptotic expansion of the Lambert series \begin{equation*} y^k \sum_{n=1}^\infty λ_{f}( n^2 ) \exp (- ny), \quad \textrm{as}\,\, y \rightarrow 0^{+}, \end{equation*} where $λ_f(n)$…
